TITLE: Is the following a valid mathematical statement?
QUESTION [8 upvotes]: For all $f:\mathbb N\to\{1,2,3,\ldots,100\}$,
 If $f$ is a one to one correspondence,
 Then $f^{-1}(2)=3$
It seems as though this should not be a valid statement, since the implication fails to remain a statement when the hypothesis becomes false. This is because when $f$ is not one to one and onto, the syntax $f^{-1}(2)$ is undefined. So, the implication on a whole becomes undefined under such a condition. Is this true?

REPLY [8 votes]: The question is more complicated then a simple ex falso quodlibet. If the sequence of letters makes sense at all, then it is true, since there is no one-to-one function $\mathbb{N}\rightarrow\{1, \ldots, 100\}$. 
However, it is not clear whether we are talking about a logical formula at all. For example $(\exists x: x\neq x)\Rightarrow (x+\forall)$ is not a valid formula, in particular it is not true, although it is of the form "false $\Rightarrow$ something".
The real problem is that $f^{-1}$ is a short-hand notation which goes beyond the realm of formal logic. We can declare that $f^{-1}(a)=b$ means that an inverse function exists and at $a$ this inverse function takes the value $b$. If we do so, then the statement becomes true. On the other hand we can view $^{-1}$ as a partial operation on the set of functions, i.e. the inverse function, if such a function exists, and gloppidyglopp otherwise. Then any statement involving $f^{-1}$ for a function $f$, which is not a bijection becomes neither true nor false, but undefined. 
In this case this looks like (and probably is) nitpicking. However, there are real world examples where such a distinction does matter: What is the meaning of the statement $\lim a_n\neq 0$, if the limit does not exist? True, false, and undefined are all valid options.
The moral of the story is probably that when writing mathematics you should try to avoid ambiguities, and when reading mathematics, you should try to get the intended meaning, even if it differs from the formal statement.